Math Marathon?

[SumitG]

Since the forum seems to tend to stagnate over the summer, I feel like I should at least try to help counteract that this year. Obviously, the best way to do that is to do some random problems. I'll start off by giving something to prove, then whoever proves it can give the next thing.
Problem 1
Prove that if [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are not both [unparseable or potentially dangerous latex formula], then

[unparseable or potentially dangerous latex formula]

[sinkokuyou]

[sinkokuyou]

Well, I guess I'll give the next one? So, this should be pretty good for a Problem 2 since it's an 8th grade question from
第十七届“希望杯”数学邀请赛 初二 第2试：
17。How many pos int solution pairs does [unparseable or potentially dangerous latex formula] have?

[SumitG]

So, we can do some simple algebra and get that the expression is equivalent to:
[unparseable or potentially dangerous latex formula]

As x and y must both be integers, we then know that
[unparseable or potentially dangerous latex formula]

This in turn from some simple algebraic manipulation gives us that
[unparseable or potentially dangerous latex formula]

Since x must be positive (and in being so implies that y is also positive), we know that
[unparseable or potentially dangerous latex formula]
Our previous condition then implies that x can be 3,6,12, or 30, and so there are 4 positive integer solution pairs.

Problem 3
Suppose that [unparseable or potentially dangerous latex formula]. Show that [unparseable or potentially dangerous latex formula] for all [unparseable or potentially dangerous latex formula].

[sinkokuyou]

[SumitG]

Hehe yes it's rather easy. Seriously though, I thought the concept was pretty much the same as the first one. Completing the square is very, very useful. Are you going to post another problem, or should I post a somewhat more difficult completing-the-square one?

[sinkokuyou]

I have one, it's a 11th grade problem:
Problem 4
Solve for x: [unparseable or potentially dangerous latex formula].

[sinkokuyou]

and btw, this problem came from 17th(2006) 希望杯 11th grade 2nd round problem #21(10 points), the original resource is here: http://aoshoo.com/t/hg2/2shi/hope2-2-17.pdf
just to show that there's nothing wrong with the problem...

[AuSmith]

[AuSmith]

[AuSmith]

So, here's a problem:

Can you cover the Cantor set with intervals of lengths [unparseable or potentially dangerous latex formula]? Give an example or a proof that there is none.

[SumitG]

[AuSmith]

To cover the Cantor set, I mean to be contained in a union... here let me formulate the question more precisely.

If we denote the Cantor set by [unparseable or potentially dangerous latex formula], does there exist a sequence of intervals [unparseable or potentially dangerous latex formula] satisfying

[unparseable or potentially dangerous latex formula]

and

[unparseable or potentially dangerous latex formula]

?

[AuSmith]

What is your first instinct? What goes wrong? Can you use the fact that C is a closed set (I.e. converging sequences in [unparseable or potentially dangerous latex formula] converge to a limit in [unparseable or potentially dangerous latex formula])?

[AuSmith]

Well, this thread kind of died. If somebody wants to post another problem, that's okay. I'll go ahead and solve this one.

Given a sequence of open intervals [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula], we can construct a sequence of numbers [unparseable or potentially dangerous latex formula] in [unparseable or potentially dangerous latex formula] in the following way:

1) [unparseable or potentially dangerous latex formula] can only intersect at most one of the halves of [unparseable or potentially dangerous latex formula], because they are separated by a distance of [unparseable or potentially dangerous latex formula] (notice that this requires that [unparseable or potentially dangerous latex formula] be open). Let [unparseable or potentially dangerous latex formula] belong to a half [unparseable or potentially dangerous latex formula] that [unparseable or potentially dangerous latex formula] does not intersect.

2) [unparseable or potentially dangerous latex formula] can only intersect at most one of the two fourths in the half that [unparseable or potentially dangerous latex formula] lies in, because they are separated by a distance of [unparseable or potentially dangerous latex formula]. Let [unparseable or potentially dangerous latex formula] belong to a fourth [unparseable or potentially dangerous latex formula] that neither [unparseable or potentially dangerous latex formula] nor [unparseable or potentially dangerous latex formula] intersect.

3) etc.

We then have a convergent sequence of real numbers [unparseable or potentially dangerous latex formula], as

[unparseable or potentially dangerous latex formula],

and the real numbers are "complete." Look it up if you don't know what it means. Also, the Cantor set [unparseable or potentially dangerous latex formula] is "closed," so you can rely on [unparseable or potentially dangerous latex formula] belonging to [unparseable or potentially dangerous latex formula].

Finally, no interval [unparseable or potentially dangerous latex formula] can contain [unparseable or potentially dangerous latex formula], because [unparseable or potentially dangerous latex formula] would be open, surrounding [unparseable or potentially dangerous latex formula] on both sides, and the sequence [unparseable or potentially dangerous latex formula] gets arbitrarily close to [unparseable or potentially dangerous latex formula]. Which means there would have to be an [unparseable or potentially dangerous latex formula] so that [unparseable or potentially dangerous latex formula]. By construction, [unparseable or potentially dangerous latex formula], so the sequence of intervals cannot cover [unparseable or potentially dangerous latex formula].

If you wanted to further the problem to closed intervals, remember that [unparseable or potentially dangerous latex formula] is uncountable and so is every branch of [unparseable or potentially dangerous latex formula] (branches of [unparseable or potentially dangerous latex formula] are scaled versions of [unparseable or potentially dangerous latex formula]). Therefore, we can use the same argument on [unparseable or potentially dangerous latex formula].